## Journal of Differential Geometry

### Bifurcation of periodic solutions to the singular Yamabe problem on spheres

#### Abstract

We obtain uncountably many periodic solutions to the singular Yamabe problem on a round sphere that blow up along a great circle. These are (complete) constant scalar curvature metrics on the complement of $\mathbb{S}^1$ inside $\mathbb{S}^m , m \geq 5$, that are conformal to the round (incomplete) metric and periodic in the sense of being invariant under a discrete group of conformal transformations. These solutions come from bifurcating branches of constant scalar curvature metrics on compact quotients of $\mathbb{S}^m \setminus \mathbb{S}^1 \cong \mathbb{S}^{m-2} \times \mathbb{H}^2$.

#### Article information

Source
J. Differential Geom., Volume 103, Number 2 (2016), 191-205.

Dates
First available in Project Euclid: 16 May 2016

https://projecteuclid.org/euclid.jdg/1463404117

Digital Object Identifier
doi:10.4310/jdg/1463404117

Mathematical Reviews number (MathSciNet)
MR3504948

Zentralblatt MATH identifier
1348.53044

#### Citation

Bettiol, Renato G.; Piccione, Paolo; Santoro, Bianca. Bifurcation of periodic solutions to the singular Yamabe problem on spheres. J. Differential Geom. 103 (2016), no. 2, 191--205. doi:10.4310/jdg/1463404117. https://projecteuclid.org/euclid.jdg/1463404117